SIMPLE QUESTION! Open interval homeomorphic to R?

  • Thread starter fleazo
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  • #1
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Main Question or Discussion Point

Hi, I am having a major brain fart.

I realize that for example, open intervals and R are all topologically equivalent.


Similarly, closed, bounded intervals are topologically equivalent


And half open intervals and closed unbounded intervals are equivalent

But I am having a difficult time coming up with actual functions. For example, what is a function that would be a homeomorphism from (-1,5) --> R ?


I would REALLY appreciate some help here as my final is tomorrow morning!

Thanks!
 

Answers and Replies

  • #2
quasar987
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First take a function that send (a,b) to ((b-a)/2, (b-a)/2) simply by translation. Then take a dilatation that inflates of shrinks that to (-pi/2, pi/2). Then apply tan.
 
  • #3
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The above suggestion seemed a little off, but I did find a function from (a,b) > (-1,1) using the Cartesian plane using slope and evaluating for the 'intercept' at -1. Then I dilated by pi/2 and stretched with tan, mapping (a,b) onto R with a composition of continuous bijections.
 
  • #4
Bacle2
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I can't tell why you think the suggestion is off; seems pretty reasonable, since size/area/volume are not topological invariants , so that rescaling does not change
the topology of a space.
 
  • #5
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SIMPLE ANSWER! Yes!
 
  • #6
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First, he's sending the interval (a,b) to ((b-a)/2, (b-a)/2), which means sending (-2,1) to (-3,-3). If anything he's missing a negative sign.
 
  • #7
mathwonk
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have you looked at the graph of tan(x) lately?
 
  • #8
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Not lately. I just checked and it looks like how I remember it. Am I missing something?
 

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